arXiv subjects

Li Ma

Publications and source records attributed to Li Ma.

252 records · Page 14Linked to original sources

arXiv2007

Properties of positive solutions of an Elliptic Equation with negative exponents

In this paper, we study the existence and non-existence result of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics. We can get some useful local gradient estimate and L1 lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.

math.AP↗

arXiv2007

Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrodinger system

In this paper, we establish two new types of invariant sets for the coupled nonlinear Schrodinger system on $\mathbb{R}^n$, and derive two sharp thresholds of blow-up and global existence for its solution. Some analogous results for the nonlinear Schrodinger system posed on the hyperbolic space $\mathbb{H}^n$ and on the standard 2-sphere $\mathbb{S}^2$ are also presented. Our arguments and constructions are improvements of some previous works on this direction. At the end, we give some heuristic analysis about the strong instability of the solitary waves.

math.AP↗

arXiv2007

L2 p-Forms and Ricci flow with bounded curvature on manifolds

In this paper, we study the evolution of L2 p-forms under Ricci flow with bounded curvature on a complete non-compact or a compact Riemannian manifold. We show that under curvature pinching conditions on such a manifold, the L2 norm of a smooth p-form is non-increasing along the Ricci flow. The L^{\infty} norm is showed to have monotonicity property too.

math.DG↗

arXiv2007

Blow-up of solutions to the nonlinear Schr$\bf{\ddot{O}}$dinger equations on standard N-sphere and hyperbolic N-space

In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr$\ddot{o}$dinger equations on some Riemannian manifolds like the standard 2-sphere $S^2$ and the hyperbolic 2-space $H^{2}(-1)$. Using the similar idea, we establish such blow-up results on higher dimensional standard sphere and hyperbolic $n$-space. Extensions to $n$-dimensional Riemannian warped product manifolds with $n\geq 2$ are also given.

math.CA↗

arXiv2005

Eigenvalue monotonicity for the Ricci-Hamilton flow, revised version

This is a revised version of our short note [arxiv.math.DG/0403065] where we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We show that the eigenvalue of the Lapacian operator on a compact domain associated with the evolving Ricci flow is non-decreasing provided the scalar curvature having a non-negative lower bound and Einstein tensor being not too negative. This result will be useful in the study of blow-up models of the Ricci-Hamilton flow. We remark that using our idea, D.Kokotov and D.Korotkin [Normalized Ricci Flow on Riemann Surfaces and Determinant of Laplacian, Letters in Mathematical Physics, 71(3)(2005)241-242] can give a simple proof of the fact that the determinant of Laplace operator in a smooth metric over compact Riemann surfaces of an arbitrary genus g monotonously grows under the normalized Ricci flow. Together with results of Hamilton and B.Chow that under the action of the normalized Ricci flow a smooth metric tends asymptotically to the metric of constant curvature, this leads to a simple proof of the Osgood-Phillips-Sarnak theorem stating that within the class of smooth metrics with fixed conformal class and fixed volume the determinant of the Laplace operator is maximal on the metric of constant curvature. One may consider similar problems for other geometric flow and other elliptic operators, such as Yamabe flow, or Kahler-Ricci flow.

math.DG↗

arXiv2005

New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems

In this paper, we establish a general monotonicity formula of the following elliptic system $$ Δu_i+f_i(u_1,...,u_m)=0 \quad {\rm in} Ω, \label{0.1} $$ where $Ω\subset\subset \mathbb{R}^n$ is a bounded domain, $(f_i(u_1,...,u_m))=\nabla F(\vec{u})$, and $F(\vec{u})$ is a given smooth function of $\vec{u}=(u_1,...,u_m)$, $m,n$ are two positive integers. We also set up a new monotonicity formula for the following parabolic system $$ \partial_t u_i-Δu_i-f_i(u_1,...,u_m)=0, in (t_1, t_2)\times \mathbb{R}^n, $$ where $t_1<t_2$ are two constants, $(f_i(\vec{u}))$ is given as above. Our new monotonicity formulae are focused on more attention to the monotonicity of non-linear terms. Our point of view is that we introduce an index called $β$ to measure the monotonicity of the non-linear terms in the problems. The index in the study of monotonicity formulae is very useful in understanding the behavior of blow up sequences of solutions. Corresponding monotonicity results for free boundary problems are also presented.

math.AP↗

arXiv2005

Mass under the Ricci flow

In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let $(M,g)$ be an ALE manifold of dimension $n=3$. If $m(g)\neq 0$, then the Ricci flow starting at $g$ can not have Euclidean space as its (uniform) limit.

math.DG↗

arXiv2005

$L^2$ Forms and Ricci flow with bounded curvature on Complete Non-compact manifolds

In this paper, we study the evolution of $L^2$ one forms under Ricci flow with bounded curvature on a non-compact Rimennian manifold. We show on such a manifold that the $L^2$ norm of a smooth one form with compact support is non-increasing along the Ricci flow with bounded curvature. The $L^{\infty}$ norm is showed to have monotonicity property too. Then we use $L^{\infty}$ cohomology of one forms with compact support to study the singularity model for the Ricci flow on $S^1\times \mathbb{R}^{n-1}$.

math.DG↗

arXiv2005

Remarks on non-compact complete Ricci expanding solitons

In this paper, we study gradient Ricci expanding solitons $(X,g)$ satisfying $$ Rc=cg+D^2f, $$ where $Rc$ is the Ricci curvature, $c<0$ is a constant, and $D^2f$ is the Hessian of the potential function $f$ on $X$. We show that for a gradient expanding soliton $(X,g)$ with non-negative Ricci curvature, the scalar curvature $R$ has at least one maximum point on $X$, which is the only minimum point of the potential function $f$. Furthermore, $R>0$ on $X$ unless $(X,g)$ is Ricci flat. We also show that there is exponentially decay for scalar curvature for $ε$-pinched complete non-compact expanding solitons.

math.DG↗

arXiv2004

Some properties of non-compact complete Riemannian manifolds

In this paper, we study the volume growth property of a non-compact complete Riemannian manifold $X$. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to study gradient Ricci solitons. We also show that on $X$, for any $q\in (0,\infty)$, every non-negative $L^q$ subharmonic function is constant under a natural decay condition on the Ricci curvature.

math.DG↗

arXiv2004

Remarks on Gradient Ricci Solitons

In this paper, we study the gradient Ricci soliton equation on a complete Riemannian manifold. We show that under a natural decay condition on the Ricci curvature, the Ricci soliton is Ricci-flat and ALE.

math.DG↗

arXiv2004

Radial Symmetry and Monotonicity Results for an Integral Equation

In this paper, we consider radial symmetry property of positive solutions of an integral equation arising from some higher order semi-linear elliptic equations on the whole space $\mathbf{R}^n$. We do not use the usual way to get symmetric result by using moving plane method. The nice thing in our argument is that we only need a Hardy-Littlewood-Sobolev type inequality. Our main result is Theorem 1 below.

math.AP↗

arXiv2004

Interior Gradient Bound For Minimal Graphs in a Product Manifold

Let $(M, g)$ be an $n$-dimensional complete Riemannian manifold with $Ric(M)\geq-(n-1)Q$, where $Q\geq0$ is a constant. We obtain an interior gradient bound for minimal graphs in $M\times R$ under some technical assumptions. For details, see Theorem 2.

math.DG↗

arXiv2004

Examples for Cross Curvature Flow on 3-Manifolds

Recently, B.Chow and R.S.Hamilton introduced the cross curvature flow on 3-manifolds. In this paper, we analyze two interesting examples for this new flow. One is on a square torus bundle over a circle, and the other is on a $S^{2}$ bundle over a circle. We show that the global flow exist in both cases. But on the former the flow diverges at time infinity, and on the latter the flow converges at time infinity.

math.DG↗

arXiv2003

Curve Shortening Flow in a Riemannian Manifold

In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the global behavior of the flow. In particular, we show the following results. 1). Let $\mathbf{M}$ be a compact locally symmetric space. If the curve shortening flow exists for infinite time, and $$ \lim_{t\to\infty}L(γ_{t})>0, $$ then for every $n>0$, $$ \lim_{t\to \infty}\sup(|\frac{D^{n}T}{\partial s^{n}}|)=0. $$ In particular, the limiting curve exists and is a closed geodesic in $\mathbf{M}$. 2). For $γ_{0}$ is a ramp, we have a global flow and the flow converges to a geodesic in $C^{\infty}$ norm.

math.DG↗

arXiv2003

Asymptotic behavior of positive solutions of some quasilinear elliptic problems

We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem $-Δ_p u=a u^{p-1}-b(x) u^q$, $u|_{\partial Ω}=0$ as $q \to p-1+0$ and as $q \to \infty$ via a scale argument. Here $Δ_p$ is the $p$-Laplacian with $1 p-1$. If $p=2$, such problems arise in population dynamics. Our main results generalize the results for $p=2$, but some technical difficulties arising from the nonlinear degenerate operator $-Δ_p$ are successfully overcome. As a by-product, we can solve a free boundary problem for a nonlinear $p$-Laplacian equation.

math.AP↗

arXiv2003

Dirichlet problems of a quasi-linear elliptic system

We discuss the Dirichlet problem of the quasi-linear elliptic system \begin{eqnarray*} -e^{-f(U)}div(e^{f(U)}\bigtriangledown U)+&{1/2}f'(U)|\bigtriangledown U|^2&=0, {in $Ω$}, & U|_{\partialΩ}&=ϕ. \end{eqnarray*} Here $Ω$ a smooth bounded domain in $R^n$, $f: R^N\to R$ is a smooth function, $U:Ω\to R^N$ is the unknown vector-valued function, $ϕ:\barΩ\to R^N$ is a given vector-valued $C^2$ function, $f'$ is the gradient of the function $f$ with respect to the variable $U$. Such problems arise in population dynamics and Differential Geometry. The difficulty of studying this problem is that this nonlinear elliptic system does not fit the usual growth condition in M.Giaquinta's book [G] and the natural working space $H^1\cap L^{\infty}(Ω)$ for the corresponding Euler-Lagrange functional does not fit the usual minimization or variational argument. We use the direct method on a convex subset of $H^1\cap L^{\infty}(Ω)$ to overcome these difficulties. Under a suitable assumption on the function $f$, we prove that there is at least one solution to this problem. We also give application of our result to the Dirichlet problem of harmonic maps into the standard sphere

math.AP↗

arXiv2003

B-sub-manifolds and their stability

In this paper, we introduce a concept of B-minimal sub-manifolds and discuss the stability of such a sub-manifold in a Riemannian manifold $(M,g)$. Assume $B(x)$ is a smooth function on $M$. By definition, we call a sub-manifold $Σ$ {\em B-minimal} in $(M,g)$ if the product sub-manifold $Σ\times S^1$ is a {\em minimal} sub-manifold in a warped product Riemannian manifold $(M\times S^1, g+e^{2B(x)}dt^2)$, so its stability is closely related to the stability of solitons of mean curvature flows as noted earlier by G. Huisken, S. Angenent, and K. Smoczyk. We can show that the "grim reaper" in the curve-shortening problem is stable in the sense of "symmetric stable" defined by K. Smoczyk. We also discuss the graphic B-minimal sub-manifold in $R^{n+k}$.

math.DG↗